Constant-sum games as a way from non-cell autonomous processes to constant tumour growth rate

A lot of thinking in cancer biology seems to be focused on cell-autonomous processes. This is the (overly) reductive view that key properties of cells, such as fitness, are intrinsic to the cells themselves and not a function of their interaction with other cells in the tumour. As far as starting points go, this is reasonable. But in many cases, we can start to go beyond this cell-autonomous starting point and consider non-cell-autonomous processes. This is when the key properties of a cell are not a function of just that cell but also its interaction partners. As an evolutionary game theorist, I am clearly partial to this view.

Recently, I was reading yet another preprint that has observed non-cell autonomous fitness in tumours. In this case, Johnson et al. (2019) spotted the Allee effect in the growth kinetics of cancer cells even at extremely low densities (seeding in vitro at <200 cells in a 1 mm^3 well). This is an interesting paper, and although not explicitly game-theoretic in its approach, I think it is worth reading for evolutionary game theorists.

Johnson et al.'s (2019) approach is not explicitly game-theoretic because they consider their in vitro populations as a monomorphic clonal line, and thus don't model interactions between types. Instead, they attribute non-cell autonomous processes to density dependence of the single type on itself. In this setting, they reasonably define the cell-autonomous null-model as constant exponential growth, i.e. for some constant fitness and total tumour size .

It might also be tempting to use the same model to capture cell-autonomous growth in game-theoretic models. But this would be mistaken. For this is only effectively cell-autonomous at the level of the whole tumour, but could hide non-cell-autonomous fitness at the level of the different types that make up the tumour. This apparent cell-autonomous total growth will happen whenever the type interactions are described by constant-sum games.

Given the importance of constant-sum games (more famously known as zero-sum games) to the classical game theory literature, I thought that I would write a quick introductory post about this correspondence between non-cell autonomous constant-sum games and effectively cell-autonomous growth at the level of the whole tumour.

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Quick introduction: Evolutionary game assay in Python

It’s been a while since I’ve shared or discussed code on TheEGG. So to avoid always being too vague and theoretical, I want to use this post to explain how one would write some Python code to measure evolutionary games. This will be an annotated sketch of the game assay from our recent work on measuring evolutionary games in non-small cell lung cancer (Kaznatcheev et al., 2019).

The motivation for this post came about a month ago when Nathan Farrokhian was asking for some advice on how to repeat our game assay with a new experimental system. He has since done so (I think) by measuring the game between Gefitinib-sensitive and Gefitinib-resistant cell types. And I thought it would make a nice post in the quick introductions series.

Of course, the details of the system don’t matter. As long as you have an array of growth rates (call them yR and yG with corresponding errors yR_e and yG_e) and initial proportions of cell types (call them xR and xG) then you could repeat the assay. To see how to get to this array from more primitive measurements, see my old post on population dynamics from time-lapse microscopy. It also has Python code for your enjoyment.

In this post, I’ll go through the two final steps of the game assay. First, I’ll show how to fit and visualize fitness functions (Figure 3 in Kaznatcheev et al., 2019). Second, I’ll transform those fitness functions into game points and plot (Figure 4b in Kaznatcheev et al., 2019). I’ll save discussions of the non-linear game assay (see Appendix F in Kaznatcheev et al., 2019) for a future post.

Local peaks and clinical resistance at negative cost

Last week, I expanded on Rob Noble’s warning about the different meanings of de novo resistance with a general discussion on the meaning of resistance in a biological vs clinical setting. In that post, I suggested that clinicians are much more comfortable than biologists with resistance without cost, or more radically: with negative cost. But I made no argument — especially no reductive argument that could potentially sway a biologist — about why we should entertain the clinician’s perspective. I want to provide a sketch for such an argument in this post.

In particular, I want to present a theoretical and extremely simple fitness landscape on which a hypothetical tumour might be evolving. The key feature of this landscape is a low local peak blocking the path to a higher local peak — a (partial) ultimate constraint on evolution. I will then consider two imaginary treatments on this landscape, one that I find to be more similar to a global chemotherapy and one that is meant to capture the essence of a targetted therapy. In the process, I will get to introduce the idea of therapy transformations to a landscape — something to address the tendency of people treating treatment fitness landscapes as completely unrelated to untreated fitness landscapes.

Of course, these hypothetical landscapes are chosen as toy models where we can have resistance emerge with a ‘negative’ cost. It is an empirical question to determine if any of this heuristic capture some important feature of real cancer landscapes.

But we won’t know until we start looking.

Effective games from spatial structure

For the last week, I’ve been at the Institute Mittag-Leffler of the Royal Swedish Academy of Sciences for their program on mathematical biology. The institute is a series of apartments and a grand mathematical library located in the suburbs of Stockholm. And the program is a mostly unstructured atmosphere — with only about 4 hours of seminars over the whole week — aimed to bring like-minded researchers together. It has been a great opportunity to reconnect with old colleagues and meet some new ones.

During my time here, I’ve been thinking a lot about effective games and the effects of spatial structure. Discussions with Philip Gerlee were particularly helpful to reinvigorate my interest in this. As part of my reflection, I revisited the Ohtsuki-Nowak (2006) transform and wanted to use this post to share a cute observation about how space can create an effective game where there is no reductive game.

Suppose you were using our recent game assay to measure an effective game, and you got the above left graph for the fitness functions of your two types. On the x-axis, you have seeding proportion of type C and on the y-axis you have fitness. In cyan you have the measured fitness function for type C and in magenta, you have the fitness function for type D. The particular fitnesses scale of the y-axis is not super important, not even the x-intercept — I’ve chosen them purely for convenience. The only important aspect is that the cyan and magenta lines are parallel, with a positive slope, and the magenta above the cyan.

This is not a crazy result to get, compare it to the fitness functions for the Alectinib + CAF condition measured in Kaznatcheev et al. (2018) which is shown at right. There, cyan is parental and magenta is resistant. The two lines of best fit aren’t parallel, but they aren’t that far off.

How would you interpret this sort of graph? Is there a game-like interaction happening there?

Of course, this is a trick question that I give away by the title and set-up. The answer will depend on if you’re asking about effective or reductive games, and what you know about the population structure. And this is the cute observation that I want to highlight.

Looking for species in cancer but finding strategies and players

Sometime before 6 August 2014, David Basanta and Tamir Epstein were discussing the increasing focus of mathematical oncology on tumour heterogeneity. An obstacle for this focus is a good definitions of heterogeneity. One path around this obstacle is to take definitions from other fields like ecology — maybe species diversity. But this path is not straightforward: we usually — with some notable and interesting examples — view cancer cells as primarily asexual and the species concept is for sexual organisms. Hence, the specific question that concerned David and Tamir: is there a concept of species that applies to cancer?

I want to consider a couple of candidate answers to this question. None of these answers will be a satisfactory definition for species in cancer. But I think the exercise is useful for understanding evolutionary game theory. With the first attempt to define species, we’ll end up using the game assay to operationalize strategies. With the second attempt, we’ll use the struggle for existence to define players. Both will be sketches that I will need to completely more carefully if there is interest.

Token vs type fitness and abstraction in evolutionary biology

There are only twenty-six letters in the English alphabet, and yet there are more than twenty-six letters in this sentence. How do we make sense of this?

Ever since I first started collaborating with David Basanta and Jacob Scott back in 2012/13, a certain tension about evolutionary games has been gnawing at me. A feeling that a couple of different concepts are being swept up under the rug of a single name.^[1] This feeling became stronger during my time at Moffitt, especially as I pushed for operationalizing evolutionary games. The measured games that I was imagining were simply not the same sort of thing as the games implemented in agent-based models. Finally this past November, as we were actually measuring the games that cancer plays, a way to make the tension clear finally crystallized for me: the difference between reductive and effective games could be linked to two different conceptions of fitness.

This showed a new door for me: philosophers of biology have already done extensive conceptual analysis of different versions of fitness. Unfortunately, due to various time pressures, I could only peak through the keyhole before rushing out my first draft on the two conceptions of evolutionary games. In particular, I didn’t connect directly to the philosophy literature and just named the underlying views of fitness after the names I’ve been giving to the games: reductive fitness and effective fitness.

Now, after a third of a year busy teaching and revising other work, I finally had a chance to open that door and read some of the philosophy literature. This has provided me with a better vocabulary and clearer categorization of fitness concepts. Instead of defining reductive vs effective fitness, the distinction I was looking for is between token fitness and type fitness. And in this post, I want to discuss that distinction. I will synthesize some of the existing work in a way that is relevant to separating reductive vs. effective games. In the process, I will highlight some missing points in the current debates. I suspect this points have been overlooked because most of the philosophers of biology are focused more on macroscopic organisms instead of the microscopic systems that motivated me.^[2]

Say what you will of birds and ornithology, but I am finding reading philosophy of biology to be extremely useful for doing ‘actual’ biology. I hope that you will, too.

Deadlock & Leader as deformations of Prisoner’s dilemma & Hawk-Dove games

Recently, I’ve been working on revisions for our paper on measuring the games that cancer plays. One of the concerns raised by the editor is that we don’t spend enough time introducing game theory and in particular the Deadlock and Leader games that we observed. This is in large part due to the fact that these are not the most exciting games and not much theoretic efforts have been spent on them in the past. In fact, none that I know of in mathematical oncology.

With that said, I think it is possible to relate the Deadlock and Leader games to more famous games like Prisoner’s dilemma and the Hawk-Dove games; both that I’ve discussed at length on TheEGG. Given that I am currently at the Lorentz Center in Leiden for a workshop on Understanding Cancer Through Evolutionary Game Theory (follow along on twitter via #cancerEGT), I thought it’d be a good time to give this description here. Maybe it’ll inspire some mathematical oncologists to play with these games.

Replicator dynamics and the simplex as a vector space

Over the years of TheEGG, I’ve chronicled a number of nice properties of the replicator equation and its wide range of applications. From a theoretical perspective, I showed how the differential version can serve as the generator for the action that is the finite difference version of replicator dynamics. And how measurements of replicator dynamics can correspond to log-odds. From an application perspective, I talked about how replicator dynamics can be realized in many different ways. This includes a correspondance to idealized replating experiments and a representation of populations growing toward carrying capacity via fictitious free-space strategies. These fictitious strategies are made apparent by using a trick to factor and nest the replicator dynamics. The same trick can also help us to use the symmetries of the fitness functions for dimensionality reduction and to prove closed orbits in the dynamics. And, of course, I discussed countless heuristic models and some abductions that use replicator dynamics.

But whenever some object becomes so familiar and easy to handle, I get worried that I am missing out on some more foundational and simple structure underlying it. In the case of replicator dynamics, Tom Leinster’s post last year on the n-Category Cafe pointed me to the simple structure that I was missing: the vector space structure of the simplex. This allows us to use linear algebra — the friendliest tool in the mathematician’s toolbox — in a new way to better understand evolutionary dynamics.

A 2-simplex with some of its 1-dimensional linear subspaces drawn by Greg Egan.

Given my interest in operationalization of replicator dynamics, I will use some of the terminology and order of presentation from Aitchison’s (1986) statistical analysis of compositional data. We will see that a number of operations that we define will have clear experimental and evolutionary interpretations.

I can’t draw any real conclusions from this, but I found it worth jotting down for later reference. If you can think of a way to make these observations useful then please let me know.

Dark selection from spatial cytokine signaling networks

Greetings, Theory, Evolution, and Games Group! It’s a pleasure to be on the other side of the keyboard today. Many thanks to Artem for the invite to write about some of our recent work and the opportunity to introduce myself via this post. I do a bit of blogging of my own over at vcannataro.com — mostly about neat science I stumble over while figuring out my way.

I’m a biologist. I study the evolutionary dynamics within somatic tissue, or, how mutations occur, compete, accumulate, and persist in our tissues, and how these dynamics manifest as aging and cancer (Cannataro et al., 2017a). I also study the evolutionary dynamics within tumors, and the evolution of resistance to targeted therapy (Cannataro et al., 2017b).

In November 2016 I attended the Integrated Mathematical Oncology Workshop on resistance, a workweek-long intensive competitive workshop where winners receive hard-earned $$ for research, and found myself placed in #teamOrange along with Artem. In my experience at said workshop (attended 2015 and 2016), things usually pan out like this: teams of a dozen or so members are assembled by the workshop organizers, insuring a healthy mix of background-education heterogeneity among groups, and then after the groups decide on a project they devise distinct but intersecting approaches to tackle the problem at hand. I bounced around a bit early on within #teamOrange contributing to our project where I could, and when the need for a spatially explicit model of cytokine diffusion and cell response came up I jumped at the opportunity to lead that endeavor. I had created spatially explicit cellular models before — such as a model of cell replacement in the intestinal crypt (Cannataro et al., 2016) — but never one that incorporated the diffusion or spread of some agent through the space. That seemed like a pretty nifty tool to add to my research kit. Fortunately, computational modeler extraordinaire David Basanta was on our team to teach me about modeling diffusion (thanks David!).

Below is a short overview of the model we devised.

Ratcheting and the Gillespie algorithm for dark selection

In Artem’s previous post about the IMO workshop he suggests that “[s]ince we are forced to move from the genetic to the epigenetic level of description, it becomes important to suggest a plausible mechanism for heritable epigenetic effects. We need to find a stochastic ratcheted phenotypic switch among the pathways of the CMML cells.” Here I’ll go into more detail about modeling this ratcheting and how to go about identifying the mechanism. We can think of this as a potential implementation of the TYK bypass in the JAK-STAT pathway described experimentally by Koppikar et al. (2012). However, I won’t go into the specifics of exact molecules, keeping to the abstract essence.

After David Robert Grime’s post on oxygen use, this is the third entry in our series on dark selection in chronic myelomonocytic leukemia (CMML). We have posted a preprint (Kaznatcheev et al., 2017) on our project to BioRxiv and section 3.1 therein follows this post closely.